1. Understand the reaction

Acylation of an amino group with acetyl chloride occurs as:

$-NH_2 + CH_3COCl \rightarrow -NHCOCH_3 + HCl$

Each amino group gets converted into an acetamide group.

2. Find mass increase per amino group

In this reaction, one hydrogen from $-NH_2$ is replaced by the acetyl group $COCH_3$.

So, net mass added per amino group is:

$M(COCH_3) - M(H) = (12+16+12+3) - 1 = 43 - 1 = 42$

Equivalently, each acylation increases molecular mass by $42$.

3. Use the given molecular masses

Initial molecular mass:

$180$

Final molecular mass after acylation:

$390$

So total increase in mass is:

$390 - 180 = 210$

4. Calculate number of amino groups

If each amino group increases mass by $42$, then number of amino groups is:

$n = \frac{210}{42} = 5$

5. Match with options

Thus, the compound contains:

$\boxed{5}$

So the correct option is B.